Answer :
Relative dimensions which will make the surface area a maximum is
x=y = \sqrt{area/3 }
z = 1/2( \sqrt{area/3 })
Volume = 2xh
Let the box have dimensions x,y,z>0.
The surface area is A=xy+2xz+2yz
The volume is V=xyz
The Lagrangian is L=V−λ(A−constant)=xyz−λ(xy+2xz+2yz−constant).
∂L∂x=yz−λ(y+2z)=0
∂L∂y=xz−λ(x+2z)=0
∂L∂z=xy−λ(2x+2y)=0
Hence, λ=yzy+2z=xzx+2z=xy2x+2y
On cross multiplication,
xyz+2yz2=xyz+2xz2which implies x=y.
Cross-multiplying the middle and the right expressions ,2x2z+2xyz=x2y+2xyzwhich implies z=12y.
So, optimal volume with fixed surface area is to have the width and the depth equal and the height to be half the width and depth.
To find x,y,z given a fixed surface area Area one would use x=y=2z in the equation
Area=xy+2xz+2yz=x2+x2+x2 so
x=y = \sqrt{area/3 }
z = 1/2( \sqrt{area/3 })
Volume is a measurement of three-dimensional space that is occupied. Numerous imperial units or SI-derived units, such as the cubic metre and litre, are frequently used to quantify it numerically (such as the gallon, quart, cubic inch). Volume and length (cubed) have a symbiotic relationship.
To learn more about surface area
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